# True Discount Questions

**FACTS AND FORMULAE FOR TRUE DISCOUNT QUESTIONS**

Suppose a man has to pay Rs.156 after 4 years and the rate of interest is 14% per annum. Clearly, Rs. 100 at 14% will amount to Rs. 156 in 4 years. So, the payment of Rs. 100 now will clear off the debt of Rs.156 due 4 years. Hence, we say that :

Sum due = Rs.156 due 4 years hence;

Present Worth (P.W) = Rs. 100;

True Discount (T.D) = (Sum due) - (P.W)=Rs. (156 - 100) = Rs. 56

We define :

** T.D = Interest on P.W**

** Amount = (P.W) + (T.D)**

Interest is reckoned on P.W and true discount is reckoned on the amount.

**IMPORTANT FORMULAE**

Let rate = R% per annum and Time = T years. Then,

**1. $P.W=\frac{100\times Amount}{100+\left(R\times T\right)}=\frac{100\times T.D}{R\times T}$**

**2.** $T.D=\frac{\left(P.W\right)\times R\times T}{100}=\frac{Amount\times R\times T}{100+\left(R\times T\right)}$

**3.** $Sum=\frac{\left(S.I\right)\times \left(T.D\right)}{\left(S.I\right)-\left(T.D\right)}$

**4.** (S.I) - (T.D )= S.I on T.D

**5.** When the sum is put at compound interest, then

$P.W=\frac{Amount}{{\left(1+{\displaystyle \frac{R}{100}}\right)}^{T}}$

A) 715 | B) 469 |

C) 400 | D) 750 |

Explanation:

Let C.P be Rs. x

900 - x = 2(x - 450) => x = Rs.600

C.P = 600 gain required is 25%

S.P = [(100+25) x 600] / 100 = Rs.750

A) gains Rs. 55 | B) gains Rs. 50 |

C) loses Rs. 30 | D) gains Rs. 30 |

A) 0% | B) 5% |

C) 7.5% | D) 10% |

Explanation:

C.P = Rs.3000

S.P =Rs. [3600 x 10] / [100+(10 x 2)] = Rs.3000

Gain =0%

A) 18.20 | B) 18.33 |

C) 18 | D) 18.30 |

Explanation:

Present worth = Amount - True Discount = 110 -10 = Rs.100

SI on Rs.100 for a certain time = Rs.10

SI on Rs.100 for doube the time = Rs.20

True Discount on Rs.120 = 120 - 100 = Rs.20

True Discount on Rs.110 = $\left(110\times \frac{20}{120}\right)$ = Rs.18.33

A) Rs.12880 | B) Rs.12000 |

C) Both are equally good | D) None of the above |

Explanation:

PW of Rs.12,880 due 8 months hence

= Rs. [12880 x 100] / [100+(18 x 8/12)] =Rs.11500

Clearly 12000 in cash is a better offer.

A) 12% | B) 13% |

C) 15% | D) 14% |

Explanation:

P.W = 2562-122 =Rs.2440

Rate = [100 x 122] / [2440 x (1/3)] =15%

A) Rs.6800 | B) Rs.6500 |

C) Rs.6000 | D) Rs.6200 |

A) Rs.948 | B) Rs.876 |

C) Rs.768 | D) Rs.658 |

Explanation:

P.W. = $\frac{100\times T.D}{R\times T}$ = $\frac{100\times 168}{14\times 2}$ = 600.

Sum = (P.W. + T.D.) = Rs. (600 + 168) = Rs. 768.